#156608

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
commutator	C₂⁴	C₂³×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 28 points:

(2,3)(6,8)(9,10,11,12)(14,17)(15,19)(21,22,24,26)(23,25,28,27);(1,2)(3,4)(5,6)(7,8)(14,17)(15,19)(21,22,24,26)(23,27,28,25);(2,3)(5,6,7,8)(13,14)(15,18)(16,17)(19,20)(21,22)(23,25)(24,26)(27,28);(1,3,4,2)(5,6)(7,8)(13,15,16,19)(14,18,17,20)(21,23,24,28)(22,25,26,27)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=efg,d^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 63804);